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The Uncertain Game Host

Q: You are in a game where the game host has a gold coin covered under one of three hats. However, the game host can choose to not put the gold coin under any hat with a probability of 10%. You open the first two hats and find no coin under it. What is the probability that there is a gold under the 3rd hat?

A: The problem clearly requires a Bayesian approach. However, it needs to be framed correctly, or else it may not be easy to grasp. Let us start with the hypothesis we want to test for \(H\). Clearly if the host has put in a gold coin, then the probability of winning is 1 as you have already opened up two hats. The new evidence here is that two hats have been opened and no coins have shown up. Also the probability of seeing such evidence is 1 if the host has not put coin under any hat. So the way to frame the sought probability is
$$
P(H|E) = \frac{P(E|H)\times P(H)}{P(E|H)\times P(H) + P(E|\neg H)\times (1 - P(H))}
$$
From the above: \(P(E|\neg H) = 1\) and \(P(E|H) = \frac{2}{3} \times \frac{1}{2} = \frac{1}{3}\). It is important to explain how we got to \(P(E|H) = \frac{1}{3}\). Given that the host has put in a coin somewhere there is a \(\frac{2}{3}\) chance that the first hat flip would reveal nothing and given this empty flip there is a \(\frac{1}{2}\) chance that the next flip would reveal nothing. Putting it all together gives
$$
P(H|E) = \frac{\frac{1}{3}\times \frac{9}{10}}{\frac{1}{3} \times \frac{9}{10} + 1 \times (1 - \frac{9}{10})}\\
= \frac{3}{4}
$$
Now that you have the answer, it is also intuitive to see why it remains high despite two empty flips. The probability that the host would insert is a coin is high to begin with.

Discovering Statistics Using R
This is a good book if you are new to statistics & probability while simultaneously getting started with a programming language. The book supports R and is written in a casual humorous way making it an easy read. Great for beginners. Some of the data on the companion website could be missing.

Linear Algebra (Dover Books on Mathematics)
An excellent book to own if you are looking to get into, or want to understand linear algebra. Please keep in mind that you need to have some basic mathematical background before you can use this book.

Linear Algebra Done Right (Undergraduate Texts in Mathematics)
A great book that exposes the method of proof as it used in Linear Algebra. This book is not for the beginner though. You do need some prior knowledge of the basics at least. It would be a good add-on to an existing course you are doing in Linear Algebra.

Follow @ProbabilityPuzIf you are looking to learn time series analysis, the following are some of the best books in time series analysis.

Introductory Time Series with R (Use R!)
This is good book to get one started on time series. A nice aspect of this book is that it has examples in R and some of the data is part of standard R packages which makes good introductory material for learning the R language too. That said this is not exactly a graduate level book, and some of the data links in the book may not be valid.

Econometrics
A great book if you are in an economics stream or want to get into it. The nice thing in the book is it tries to bring out a oneness in all the methods used. Econ majors need to be up-to speed on the grounding mathematics for time series analysis to use this book. Outside of those prerequisites, this is one of the best books on econometrics and time series analysis.